Certain results on invariant submanifolds of an almost Kenmotsu $(\kappa ,\mu ,\nu )$-space

Abstract

In the present paper, we study invariant submanifolds of almost Kenmotsu structures whose Riemannian curvature tensor has $(\kappa ,\mu ,\nu )$-nullity distribution. Since the geometry of an invariant submanifold inherits almost all properties of the ambient manifold, we research how the functions $\kappa ,\mu$ and $\nu$ behave on the submanifold. In this connection, necessary and sufficient conditions are investigated for an invariant submanifold of an almost Kenmotsu $(\kappa ,\mu ,\nu )$-space to be totally geodesic under some conditions.

1 Introduction

It is well known that a $(2n+1)$-dimensional contact metric manifold $\widetilde{M}$ admits an almost contact metric structure $(\phi ,\xi ,\eta ,g)$, i.e., it admits a global vector field $\xi$, called the characteristic vector field or the Reeb vector field, its dual is $\eta$, a tensor $\phi$ of type (1, 1) and the Riemannian metric tensor g such that

$$\begin{aligned} \phi ^2X=-X+\eta (X)\xi , \ \ \eta (\xi )=1,\ \ \eta \circ \phi =0, \end{aligned}$$

(1)

and

$$\begin{aligned} g(\phi {X},\phi {Y})=g(X,Y)-\eta (X)\eta (Y), \end{aligned}$$

(2)

for all $X,Y\in \Gamma (T\widetilde{M})$, where $\Gamma (T\widetilde{M})$ denotes the set of differentiable vector fields on $\widetilde{M}$ [2].

The manifold $\widetilde{M}$ together with the structure tensor $(\phi ,\xi ,\eta ,g)$ is called a contact metric manifold and we will denote it by $\widetilde{M}^{(2n+1)}(\phi ,\xi ,\eta ,g)$ in the rest of this paper.

By $\widetilde{\nabla }$, we denote the Levi-Civita connection of g, then the Riemannian curvature tensor of $\widetilde{R}$ of $\widetilde{M}^{2n+1}(\phi ,\xi ,\eta ,g)$ is given by

$$\begin{aligned} \widetilde{R}(X,Y)=\widetilde{\nabla }_X\widetilde{\nabla }_Y-\widetilde{\nabla }_Y\widetilde{\nabla }_X-\widetilde{\nabla }_{[X,Y]}, \end{aligned}$$

for all $X,Y\in \Gamma (T\widetilde{M})$.

On the other hand, we define the tensor field (1,1)-type by h

$$\begin{aligned} 2hX=(\ell _{\xi }\phi )X, \end{aligned}$$

for all $X\in \Gamma (T\widetilde{M})$, where $\ell _{\xi }$ is the Lie-derivative in the direction of $\xi$. Then the tensor field h is self-adjoint and satisfies

$$\begin{aligned} \phi {h}+h\phi =0, \ \ tr{h}=tr\phi {h}=0, \ \ h\xi =0. \end{aligned}$$

(3)

We have also these formulas for a contact metric manifold

$$\begin{aligned} \widetilde{\nabla }_X\xi =\phi {X}-\phi {h}X,\ \ \widetilde{\nabla }_\xi \phi =0. \end{aligned}$$

(4)

A contact metric manifold for which $\xi$ is Killing vector field is called a K-contact manifold. It is well known that a contact metric manifold is K-contact iff $h=0$.

The $(\kappa ,\mu )$-nullity distribution of a contact metric manifold $\widetilde{M}^{2n+1}(\phi ,\xi ,\eta ,g)$ for the pair $(\kappa ,\mu )\in \mathbb {R}^2$ is distribution

$$\begin{aligned}&\widetilde{M}(\kappa ,\mu ):p\longrightarrow {\widetilde{M}_p}(\kappa ,\mu )=\{Z_p\in {T}_{\widetilde{M}}(p):\\&\widetilde{R}(X,Y)Z=\kappa \{g(Y,Z)X-g(X,Z)Y\}+\mu \{g(Y,Z)hX-g(X,Z)hY\}\}, \end{aligned}$$

for all $X,Y\in \Gamma (T\widetilde{M})$. So if the characteristic vector field $\xi$ belongs to the $(\kappa ,\mu )$-nullity distribution, then

$$\begin{aligned} \widetilde{R}(X,Y)\xi =\kappa \{\eta (Y)X-\eta (X)Y\}+\mu \{\eta (Y)hX-\eta (X)hY\}, \end{aligned}$$

and the manifold $\widetilde{M}^{2n+1}(\phi ,\xi ,\eta ,g)$ is called $(\kappa ,\mu )$-contact metric manifold. If $\kappa$ and $\mu$ are non-constant smooth functions on $\widetilde{M}$, then the manifold $\widetilde{M}^{2n+1}(\phi ,\xi ,\eta ,g)$ is called generalized $(\kappa ,\mu )$-contact metric manifold [3].

Going beyond generalized $(\kappa ,\mu )$-space, T. Koufogiorgos, M. Markellos and V. J. Papantoniou introduced in [2] the notion of $(\kappa ,\mu ,\nu )$-contact metric manifold, its Riemannian curvature tensor $\widetilde{R}$ is given by

$$\begin{aligned} \widetilde{R}(X,Y)\xi= & {} \kappa \{\eta (Y)X-\eta (X)Y\}+\mu \{\eta (Y)hX-\eta (X)hY\}\nonumber \\&+\nu \{\eta (Y)\phi {h}X-\eta (X)\phi {h}Y\}, \end{aligned}$$

(5)

for all $X,Y\in \Gamma (T\widetilde{M})$, where $\kappa ,\mu ,\nu$ are smooth functions on $\widetilde{M}^{2n+1}$.

It is well known that an almost contact metric manifold is an almost Kenmotsu if $d\eta =0$ and $d\Phi =2\eta \Lambda \Phi$, where $\Phi (X,Y)=g(X,\phi {Y})$ is the fundamental 2-form of $\widetilde{M}^{2n+1}$. If an almost Kenmotsu manifold $\widetilde{M}^{2n+1}(\phi ,\eta ,\xi ,g)$ has a $(\kappa ,\mu ,\nu )$-nullity distribution, it is called an almost Kenmotsu $(\kappa ,\mu ,\nu )$-space [5].

Proposition 1.1

Given $\widetilde{M}^{2n+1}(\phi ,\eta ,\xi ,g)$ an almost Kenmotsu $(\kappa ,\mu ,\nu )$-space, then

$$\begin{aligned} h^2= & {} (\kappa +1)\phi ^2, \ \ \kappa \le {-1}, \end{aligned}$$

(6)

$$\begin{aligned} \xi (\kappa )= & {} 2(\kappa +1)(\nu -2) \end{aligned}$$

(7)

$$\begin{aligned} (\widetilde{\nabla }_X\phi )Y= & {} g(\phi {X}+hX,Y)\xi -\eta (Y)(\phi {X}+hX) \end{aligned}$$

(8)

$$\begin{aligned} \widetilde{\nabla }_X\xi= & {} -\phi ^2X-\phi {h}X \end{aligned}$$

(9)

$$\begin{aligned} S(X,\xi )= & {} 2n\kappa \eta (X) \end{aligned}$$

(10)

$$\begin{aligned} \widetilde{R}(\xi ,X)Y= & {} \kappa \{g(X,Y)\xi -\eta (Y)X\}+\mu \{g(hX,Y)\xi -\eta (Y)hX\}\nonumber \\&+\nu \{g(\phi {h}X,Y)\xi -\eta (Y)\phi {h}X\}. \end{aligned}$$

(11)

They proved that this type of manifold is intrinsically related to the harmonicity of the Reeb vector on contact metric 3-manifolds. Some authors have studied manifolds satisfying condition (5) but a non-contact metric structure. In this connection, P. Dacko and Z. Olszak defined an almost cosymplectic $(\kappa ,\mu ,\nu )$-space as an almost cosymplectic manifold that satisfies (5), but with $\kappa ,\mu$ and $\nu$ functions varying exclusively in the direction of $\xi$ in [6]. Later examples have been given for this type of manifold [7].

In modern analysis, the geometry of submanifolds has become a subject of growing interest for its significant applications in applied mathematics and theoretical physics. For instance, the notion of invariant submanifold is used to discuss properties of non-linear autonomous systems. Also, the notion of geodesic plays an important role in the theory of relativity. For totally geodesic submanifolds, the geodesics of the ambient manifolds remain geodesics in the submanifolds. Hence, totally geodesic submanifolds have also importance in mathematics as well as in physical sciences. There have been several papers on contact metric manifolds which admit covector field $\xi$ tangent to the submanifold. In this connection, we refer to [11, 12, 14, 16].

Now, in this part of the study, we are especially interested in an invariant submanifold of an almost Kenmotsu $(\kappa ,\mu ,\nu )$-space to be pseudoparallel, 2-pseudoparallel, Ricci-generalized pseudoparallel and 2-generalized pseudoparallel.

Pseudoparallel submanifolds have been studied in different structures and working on [8,9,10]. On the other hand, the study of the geometry of invariant submanifolds was introduced by Bejancu and Papaghuic [10]. In general, the geometry of an invariant submanifold inherits almost all properties of the ambient manifold.

In the present paper, we generalize the ambient space and investigate the conditions under which invariant pseudoparallel submanifolds of an almost Kenmotsu $(\kappa ,\mu ,\nu )$-space are totally geodesic.

Now, let M be an immersed submanifold of an almost Kenmotsu $(\kappa ,\mu ,\nu )$-space $\widetilde{M}^{2n+1}$. By $\Gamma (TM)$ and $\Gamma (T^\bot {M})$, we denote the tangent and normal subspaces of M in $\widetilde{M}$. Then the Gauss and Weingarten formulae are, respectively, given by

$$\begin{aligned} \widetilde{\nabla }_XY=\nabla _XY+\sigma (X,Y), \end{aligned}$$

(12)

and

$$\begin{aligned} \widetilde{\nabla }_XV=-A_VX+\nabla ^\bot _XV, \end{aligned}$$

(13)

for all $X,Y\in \Gamma (TM)$ and $V\in \Gamma (T^\bot {M})$, where $\nabla$ and $\nabla ^\bot$ are the induced connections on M and $\Gamma (T^\bot {M})$ and $\sigma$ and A are called the second fundamental form and shape operator of M, respectively, $\Gamma (TM)$ denotes the set differentiable vector fields on M. They are related by

$$\begin{aligned} g(A_VX,Y)=g(\sigma (X,Y),V). \end{aligned}$$

(14)

The first covariant derivative of the second fundamental form $\sigma$ is defined by

$$\begin{aligned} (\widetilde{\nabla }_X\sigma )(Y,Z)=\nabla ^\bot _X\sigma (Y,Z)-\sigma (\nabla _XY,Z)-\sigma (Y,\nabla _XZ), \end{aligned}$$

(15)

for all $X,Y,Z\in \Gamma (TM)$. If $\widetilde{\nabla }\sigma =0$, then the submanifold is said to be its second fundamental form is parallel.

By R, we denote the Riemannian curvature tensor of the submanifold M, we have the following Gauss equation

$$\begin{aligned} \widetilde{R}(X,Y)Z= & {} R(X,Y)Z+A_{\sigma (X,Z)}Y-A_{\sigma (Y,Z)}X+(\widetilde{\nabla }_X\sigma )(Y,Z)\nonumber \\&-(\widetilde{\nabla }_Y\sigma )(X,Z), \end{aligned}$$

(16)

for all $X,Y,Z\in \Gamma (TM)$.

$\widetilde{R}\cdot \sigma$ is given by

$$\begin{aligned} (\widetilde{R}(X,Y)\cdot \sigma )(U,V)= & {} R^\bot (X,Y)\sigma (U,V)-\sigma (R(X,Y)U,V)\nonumber \\&-\sigma (U,R(X,Y)V), \end{aligned}$$

(17)

where the Riemannian curvature tensor of normal bundle $\Gamma (T^\bot {M})$ is given

$$\begin{aligned} R^\bot (X,Y)=[\nabla ^\bot _X,\nabla ^\bot _Y]-\nabla ^\bot _{[X,Y]} \end{aligned}$$

On the other hand, the concircular curvature tensor for Riemannian manifold $(M^{2n+1},g)$ is given by

$$\begin{aligned} \mathcal {C}(X,Y)Z= & {} \widetilde{R}(X,Y)Z\nonumber \\&-\frac{\tau }{2n(2n+1)}\{g(Y,Z)X-g(X,Z)Y\}, \end{aligned}$$

(18)

where $\tau$ denotes the scalar curvature of M.

Similarly, the tensor $\mathcal {C}\cdot \sigma$ is defined by

$$\begin{aligned} (\mathcal {C}(X,Y)\cdot \sigma )(U,V)= & {} R^\bot (X,Y)\sigma (U,V)-\sigma (\mathcal {C}(X,Y)U,V)\nonumber \\&-\sigma (U,\mathcal {C}(X,Y)V), \end{aligned}$$

(19)

for all $X,Y,U,V\in \Gamma (TM)$.

For a (0, k)-type tensor field T, $k\ge {1}$ and a (0, 2)-type tensor field A on a Riemannian manifold (M, g), Q(A, T)-tensor field is defined by

$$\begin{aligned} Q(A,T)(X_1,X_2,\ldots ,X_k;X,Y)= & {} -T((X\wedge _AY)X_1,X_2,\ldots ,X_k)\nonumber \\&\ldots -T(X_1,X_2,\ldots ,X_{k-1},(X\wedge _AY)X_k),, \end{aligned}$$

(20)

for all $X_1,X_2,\ldots ,X_k,X,Y\in \Gamma (TM)$ [8], where

$$\begin{aligned} (X\wedge _AY)Z=A(Y,Z)X-A(X,Z)Y. \end{aligned}$$

(21)

Definition 1.2

Let M be a submanifold of a Riemannian manifold $(\widetilde{M},g)$. If there exist functions $L_1,L_2,L_3$ and $L_4$ on $\widetilde{M}$ such that

$$\begin{aligned} \widetilde{R}\cdot \sigma= & {} L_1Q(g,\sigma ), \end{aligned}$$

(22)

$$\begin{aligned} \widetilde{R}\cdot \widetilde{\nabla }\sigma= & {} L_2Q(g,\widetilde{\nabla }\sigma ), \end{aligned}$$

(23)

$$\begin{aligned} \widetilde{R}\cdot \sigma= & {} L_3Q(S,\sigma ) \end{aligned}$$

(24)

$$\begin{aligned} \widetilde{R}\cdot \widetilde{\nabla }\sigma= & {} L_4Q(S,\widetilde{\nabla }\sigma ), \end{aligned}$$

(25)

then M is, respectively, pseudoparallel, 2-pseudoparallel, Ricci-generalized pseudoparallel and 2-Ricci-generalized pseudoparallel submanifold. In particular, if $L_1=0$(resp., $L_2=0$), then M is said to be semiparallel(resp. 2-semiparallel) [9].

Kowalczyk studied the semi-Riemannian manifolds satisfying $Q(S,R)=0$ and Q(S, g)=0 [17]. Also, De and Majhi investigated the invariant submanifolds of Kenmotsu manifolds and showed that geometric conditions of invariant submanifolds of Kenmotsu manifolds are totally geodesic [13]. Recently, Hu and Wang obtained the geometric conditions of invariant submanifolds of a trans-Sasakian manifold to be totally geodesic [15]. Furthermore, the geometry of invariant submanifolds of different manifolds was studied by many geometers [8, 9, 11,12,13,14,15,16].

Motivated by the above studies, we make an attempt to study the invariant submanifolds of an almost Kenmotsu $(\kappa ,\mu ,\nu )$-space satisfying some geometric conditions such as $\widetilde{R}\cdot \sigma =L_1Q(g,\sigma )$, $\widetilde{R}\cdot \widetilde{\nabla }\sigma =L_2Q(g,\widetilde{\nabla }\sigma )$, $\widetilde{R}\cdot \sigma =L_3Q(S,\sigma )$ and $\widetilde{R}\cdot \widetilde{\nabla }\sigma =L_4Q(S,\widetilde{\nabla }\sigma )$.

Finally, we show that the submanifold is either totally geodesic or the functions $L_i,\kappa ,\mu$ and $\nu$ functions are restricted.

2 Invariant Submanifolds of an almost Kenmotsu $(\kappa ,\mu ,\nu )$-Space

Now, let $\widetilde{M}^{2n+1}(\phi ,\xi ,\eta ,g)$ be an almost Kenmotsu $(\kappa ,\mu ,\nu )$-space and M be an immersed submanifold of $\widetilde{M}^{2n+1}$. If $\phi (T_xM)\subseteq {T_xM}$, for each point at $x\in {M}$, then M is said to be an invariant submanifold of $\widetilde{M}^{2n+1}(\phi ,\xi ,\eta ,g)$ with respect to $\phi$. From (3), one can easily see that an invariant submanifold with respect to $\phi$ is also invariant with respect to h.

Proposition 2.1

Let M be an invariant submanifold of an almost Kenmotsu $(\kappa ,\mu ,\nu )$-space $\widetilde{M}^{2n+1}(\phi ,\xi ,\eta ,g)$ such that $\xi$ is tangent to M. Then the following equalities hold on M;

$$\begin{aligned} R(X,Y)\xi= & {} \kappa [\eta (Y)X-\eta (X)Y]+\mu [\eta (Y)hX-\eta (X)hY]\nonumber \\&+\nu [\eta (Y)\phi {h}X-\eta (X)\phi {h}Y] \end{aligned}$$

(26)

$$\begin{aligned} (\nabla _X\phi )Y= & {} g(\phi {X}+hX,Y)\xi -\eta (Y)(\phi {X}+hX) \end{aligned}$$

(27)

$$\begin{aligned} \nabla _X\xi= & {} -\phi ^2X-\phi {h}X \end{aligned}$$

(28)

$$\begin{aligned} \phi \sigma (X,Y)= & {} \sigma (\phi {X},Y)=\sigma (X,\phi {Y}), \ \ \sigma (X,\xi )=0, \end{aligned}$$

(29)

where $\nabla$, $\sigma$ and R denote the induced Levi-Civita connection on M, the shape operator and Riemannian curvature tensor of M, respectively.

Proof

We omit the proof as it is a result of direct calculations. $\square$

In the rest of this paper, we will assume that M is an invariant submanifold of an almost Kenmotsu $(\kappa ,\mu ,\nu )$-space $\widetilde{M}^{2n+1}(\varphi ,\xi ,\eta ,g)$. In this case, from (5), we have

$$\begin{aligned} \varphi {h}X=-h\varphi {X}, \end{aligned}$$

(30)

for all $X\in \Gamma (TM)$, that is, M is also invariant with respect to the tensor field h.

We need the following lemma to guarantee that the second fundamental form $\sigma$ is not always identically zero.

Lemma 2.2

Let M be an invariant submanifold of an almost Kenmotsu $(\kappa ,\mu ,\nu )$-space $M^{2n+1}(\phi ,\xi ,\eta ,g)$. Then the second fundamental form $\sigma$ of M is parallel if and only if M is totally geodesic provided $\kappa \ne {0}$.

Proof

Let us suppose that $\sigma$ is parallel. From (16), we have

$$\begin{aligned} (\widetilde{\nabla }_X\sigma )(Y,Z)=\nabla ^\bot _X\sigma (Y,Z)-\sigma (\nabla _XY,Z)-\sigma (Y,\nabla _XZ)=0, \end{aligned}$$

(31)

for all vector fields X, Y and Z on $M^{2n+1}$. Setting $Z=\xi$ in (31) and taking into account (28) and (29), we have

$$\begin{aligned} \sigma (\nabla _X\xi ,Y)=-\sigma (\phi ^2X+\phi {h}X,Y)=0, \end{aligned}$$

that is,

$$\begin{aligned} \sigma (X,Y)-\phi \sigma (hX,Y)=0. \end{aligned}$$

(32)

Writing hX of X in (30) and using (7) and (27), we obtain

$$\begin{aligned}&\sigma (hX,Y)-\phi \sigma (h^2X,Y)=0,\nonumber \\&\sigma (hX,Y)+(1+\kappa )\phi \sigma (X,Y)=0. \end{aligned}$$

(33)

From (32) and (33), we conclude that $\kappa \sigma (X,Y)=0$, which proves our assertion. The converse is obvious. $\square$

Theorem 2.3

Let M be an invariant submanifold of an almost Kenmotsu $(\kappa ,\mu ,\nu )$-space $\widetilde{M}^{2n+1}(\phi ,\eta ,\xi ,g)$. If M is a pseudoparallel submanifold of an almost Kenmotsu $(\kappa ,\mu ,\nu )$-space $\widetilde{M}^{2n+1}(\phi ,\eta ,\xi ,g)$, then M is either totally geodesic or the function $L_1$ satisfies

$$\begin{aligned} L_1=\kappa \mp \sqrt{(\kappa +1)(\nu ^2-\mu ^2)}, \ \ \mu .\nu (\kappa +1)=0. \end{aligned}$$

(34)

Proof

Let M be an invariant pseudoparallel submanifold of an almost Kenmotsu $(\kappa ,\mu ,\nu )$-space $\widetilde{M}^{2n+1}(\phi ,\eta ,\xi ,g)$. That means

$$\begin{aligned} L_1Q(g,\sigma )(U,V;X,Y)=(\widetilde{R}(X,Y)\cdot \sigma )(U,V), \end{aligned}$$

for all $X,Y,U,V\in \Gamma (TM)$, which implies that

$$\begin{aligned}&-L_1\{\sigma ((X\wedge _gY)U,V)+\sigma (U,(X\wedge _gY)V)\}=R^\bot (X,Y)\sigma (U,V)\nonumber \\&\quad -\sigma (R(X,Y)U,V)-\sigma (U,R(X,Y)V). \end{aligned}$$

Substituting $X=U=\xi$ in the last equality, and taking into account Proposition 2.1, we have

$$\begin{aligned} L_1\sigma (Y,V)=-\sigma (R(\xi ,Y)\xi ,V)=\kappa \sigma (Y,V)+\mu \sigma (hY,V)+\nu \sigma (\phi {h}Y,V), \end{aligned}$$

that is,

$$\begin{aligned} (L_1-\kappa )\sigma (V,Y)=\mu \sigma (hY,V)+\nu \phi \sigma (hY,V). \end{aligned}$$

(35)

Substituting hY for Y in (35), by view of (6) and (29), we have

$$\begin{aligned} (L_1-\kappa )\sigma (hY,V)= & {} \mu \sigma (h^2Y,V)+\nu \phi \sigma (h^2Y,V)\nonumber \\= & {} -(\kappa +1)[\mu \sigma (Y,V)+\nu \phi \sigma (Y,V)]. \end{aligned}$$

(36)

From (35) and (36), one can easily see that

$$\begin{aligned}{}[(\kappa +1)(\mu ^2-\nu ^2)+(L_1-\kappa )^2]\sigma (Y,V)+2(\kappa +1)\mu \nu \phi \sigma (Y,V)=0. \end{aligned}$$

This tell us that M is either totally geodesic submanifold or

$$\begin{aligned} (\kappa +1)(\mu ^2-\nu ^2)+(L_1-\kappa )^2=\mu \nu (\kappa +1)=0, \end{aligned}$$

(37)

which is equivalent to (34). $\square$

From Theorem 2.3, we have the following corollary.

Corollary 2.4

Let M be an invariant submanifold of an almost Kenmotsu $(\kappa ,\mu ,\nu )$-space $\widetilde{M}^{2n+1}(\phi ,\eta ,\xi ,g)$. M is a semiparallel submanifold if and only if M is totally geodesic provided

$$\begin{aligned} (\kappa +1)(\mu ^2-\nu ^2)+\kappa ^2\ne {0} \ \ or \ \ \mu \nu (\kappa +1)\ne {0}. \end{aligned}$$

Theorem 2.5

Let M be an invariant submanifold of an almost Kenmotsu $(\kappa ,\mu ,\nu )$-space $\widetilde{M}^{2n+1}(\phi ,\eta ,\xi ,g)$. If M is a Ricci-generalized pseudoparallel submanifold of an almost Kenmotsu $(\kappa ,\mu ,\nu )$-space $\widetilde{M}^{2n+1}(\phi ,\eta ,\xi ,g)$, then M is either totally geodesic or the function $L_3$ satisfies

$$\begin{aligned} L_3=\frac{1}{2n}\left[ 1\mp \frac{1}{\kappa }\sqrt{(\kappa +1)(\nu ^2-\mu ^2)}\right] , \ \ \mu \cdot \nu =0. \end{aligned}$$

(38)

Proof

Since M is an invariant Ricci-generalized pseudoparallel, there exists a function $L_3$ on M such that

$$\begin{aligned} (\widetilde{R}(X,Y)\cdot \sigma )(U,V)=L_3Q(S,\sigma )(U,V;X,Y), \end{aligned}$$

for all $X,Y,U,V\in \Gamma (TM)$. This yields to

$$\begin{aligned}&- \, L_3\{\sigma ((X\wedge _SY)U,V)+\sigma (U,(X\wedge _SY)V)\}=R^\bot (X,Y)\sigma (U,V)\nonumber \\&- \,\sigma (R(X,Y)U,V)-\sigma (U,R(X,Y)V). \end{aligned}$$

(39)

Expanding by (39) and inserting $X=V=\xi$, making use of (29) and Proposition 2.1, we obtain

$$\begin{aligned} 2n\kappa {L_3}\sigma (U,Y)=-\sigma (R(\xi ,Y)\xi ,U)= & {} \kappa \sigma (Y,U)+\mu \sigma (hY,U)\nonumber \\&+\nu \phi \sigma (hY,U), \end{aligned}$$

that is,

$$\begin{aligned} \kappa (2nL_3-1)\sigma (U,Y)=\mu \sigma (hY,U)+\nu \phi \sigma (hY,U). \end{aligned}$$

(40)

If hY is taken instead of Y in (40) and by virtue of (6) and Proposition 2.1, we reach at

$$\begin{aligned} \kappa (2nL_3-1)\sigma (U,hY)=-(\kappa +1)[\mu \sigma (hY,U)+\nu \phi \sigma (hY,U)]. \end{aligned}$$

(41)

From (40) and (41), we conclude that

$$\begin{aligned}{}[\kappa ^2(2nL_3-1)^2+(\kappa +1)(\mu ^2-\nu ^2)]\sigma (U,V)+2\mu \nu \phi \sigma (U,V)=0. \end{aligned}$$

This completes the proof. $\square$

Theorem 2.6

Let M be an invariant submanifold of an almost Kenmotsu $(\kappa ,\mu ,\nu )$-space $\widetilde{M}^{2n+1}(\phi ,\eta ,\xi ,g)$. If M is a 2-pseudoparallel submanifold of $\widetilde{M}^{2n+1}(\phi ,\eta ,\xi ,g)$, then M is either totally geodesic or the function $L_2$ satisfies the condition

$$\begin{aligned} L_2=\kappa \mp \sqrt{(\kappa +1)(\nu ^2-\mu ^2)}, \ \ \mu \nu (\kappa +1)=0. \end{aligned}$$

(42)

Proof

If M is an invariant 2-pseudoparallel of an almost Kenmotsu $(\kappa ,\mu ,\nu )$-space $\widetilde{M}^{2n+1}(\phi ,\eta ,\xi ,g)$, then there exits a function $L_2$ such that

$$\begin{aligned} L_2Q(g,\widetilde{\nabla }\sigma )(U,V,Z;X,Y)=(\widetilde{R}(X,Y)\cdot \widetilde{\nabla }\sigma )(U,V,Z), \end{aligned}$$

for all $X,Y,U,V,Z\in \Gamma (TM)$, which implies that

$$\begin{aligned}&-\, L_2\{(\widetilde{\nabla }_{(X\wedge _gY)U}\sigma )(V,Z)+(\widetilde{\nabla }_U\sigma )((X\wedge _gY)V,Z)\nonumber \\&+\, (\widetilde{\nabla }_U\sigma )(V,(X\wedge _gY)Z\}=R^\bot (X,Y)(\widetilde{\nabla }_U\sigma )(V,Z)\nonumber \\&-\, (\widetilde{\nabla }_{R(X,Y)U}\sigma )(V,Z)-(\widetilde{\nabla }_U\sigma )(R(X,Y)V,Z)\nonumber \\&-\, (\widetilde{\nabla }_U\sigma )(V,R(X,Y)Z). \end{aligned}$$

(43)

Here, substituting $\xi$ for $X=V$ in (43), we have

$$\begin{aligned}&-L_2\{(\widetilde{\nabla }_{(\xi \wedge _gY)U}\sigma )(\xi ,Z)+(\widetilde{\nabla }_U\sigma )((\xi \wedge _gY)\xi ,Z)+(\widetilde{\nabla }_U\sigma )(\xi ,(\xi \wedge _gY)Z\}\nonumber \\&\qquad \quad \quad =R^\bot (\xi ,Y)(\widetilde{\nabla }_U\sigma )(\xi ,Z)-(\widetilde{\nabla }_{R(\xi ,Y)U}\sigma )(\xi ,Z)\nonumber \\&\qquad \qquad \qquad -(\widetilde{\nabla }_U\sigma )(R(\xi ,Y)\xi ,Z)-(\widetilde{\nabla }_U\sigma )(\xi ,R(\xi ,Y)Z). \end{aligned}$$

(44)

Now we will calculate them separately. In view of (15), (21), (28) and (29), we can derive

$$\begin{aligned} (\widetilde{\nabla }_{(\xi \wedge _gY)U}\sigma )(\xi ,Z)= & {} -\sigma (\nabla _{(\xi \wedge _gY)U}\xi ,Z)\nonumber \\= & {} \sigma (\phi ^2(\xi \wedge _gY)U+\phi {h}(\xi \wedge _gY)U,Z)\nonumber \\= & {} -\sigma (g(Y,U)\xi -\eta (U)Y,Z)-\eta (U)\sigma (\phi {h}Y,Z)\nonumber \\= & {} \eta (U)\{\sigma (Y,Z)-\sigma (\phi {h}Y,Z)\}. \end{aligned}$$

(45)

In the same way,

$$\begin{aligned} (\widetilde{\nabla }_U\sigma )((\xi \wedge _gY)\xi ,Z)= & {} (\widetilde{\nabla }_U\sigma )(\eta (Y)\xi -Y,Z)\nonumber \\= & {} -\sigma (\nabla _U\eta (Y)\xi ,Z)-(\widetilde{\nabla }_U\sigma )(Y,Z)\nonumber \\= & {} -\sigma (U[\eta (Y)]\xi +\eta (Y)\nabla _U\xi ,Z)-(\widetilde{\nabla }_U\sigma )(Y,Z)\nonumber \\= & {} \eta (Y)\sigma (\phi ^2U+\phi {h}U,Z)-(\widetilde{\nabla }_U\sigma )(Y,Z)\nonumber \\= & {} \eta (Y)\{\sigma (\phi {h}U,Z)-\sigma (U,Z)\}-(\widetilde{\nabla }_U\sigma )(Y,Z), \end{aligned}$$

(46)

$$\begin{aligned} (\widetilde{\nabla }_U\sigma )(\xi ,(\xi \wedge _gY)Z)= & {} -\sigma (\nabla _U\xi ,(\xi \wedge _gY)Z)\nonumber \\= & {} \sigma (\phi ^2U+\phi {h}U,g(Y,Z)\xi -\eta (Z)Y)\nonumber \\= & {} \eta (Z)\{\sigma (U,Y)-\sigma (\phi {h}U,Y)\}. \end{aligned}$$

(47)

For the right hand side of (44), by view of Proposition 2.1, (15) and (17), we obtain

$$\begin{aligned} R^\bot (\xi ,Y)(\widetilde{\nabla }_U\sigma )(\xi ,Z)= & {} R^\bot (\xi ,Y)\{\nabla ^\bot _U\sigma (\xi ,Z)-\sigma (\nabla _U\xi ,Z)-\sigma (\xi ,\nabla _UZ)\}\nonumber \\= & {} -R^\bot (\xi ,Y)\sigma (\nabla _U\xi ,Z)=R^\bot (\xi ,Y)\sigma (\phi ^2U+\phi {h}U,Z)\nonumber \\= & {} R^\bot (\xi ,Y)\{\sigma (\phi {h}U,Z)-\sigma (U,Z)\}. \end{aligned}$$

(48)

Also, making use of (3) and (26), we derive

$$\begin{aligned} (\widetilde{\nabla }_{R(\xi ,Y)U}\sigma )(\xi ,Z)= & {} -\sigma (\nabla _{R(\xi ,Y)U}\xi ,Z)=\sigma (\phi ^2R(\xi ,Y)U+\phi {h}R(\xi ,Y)U,Z)\nonumber \\= & {} -\sigma (-\kappa \eta (U)Y-\mu \eta (U)hY-\nu \eta (U)\phi {h}Y,Z)\nonumber \\&+\sigma (-\kappa \eta (U)\phi {h}Y-\mu \eta (U)\phi {h}^2Y-\nu \eta (U)\phi {h}\phi {h}Y,Z)\nonumber \\= & {} \eta (U)\{\kappa \sigma (Y,Z)+\mu \sigma (hY,Z)+\nu \phi \sigma (hY,Z)\nonumber \\&-\kappa \phi \sigma (hY,Z)+(\kappa +1)\mu \phi \sigma (Y,Z)+\nu (\kappa +1)\sigma (Y,Z)\}, \end{aligned}$$

(49)

$$\begin{aligned} (\widetilde{\nabla }_U\sigma )(R(\xi ,Y)\xi ,Z)&\nonumber \\= & {} (\widetilde{\nabla }_U\sigma )(\kappa [\eta (Y)\xi -Y]-\mu {h}Y-\nu \phi {h}Y,Z). \end{aligned}$$

(50)

Finally,

$$\begin{aligned} (\widetilde{\nabla }_U\sigma )(\xi ,R(\xi ,Y)Z)= & {} -\sigma (\nabla _U\xi ,R(\xi ,Y)Z)=\sigma (\phi ^2U+\phi {h}U,R(\xi ,Y)Z)\nonumber \\= & {} -\sigma (-\kappa \eta (Z)Y-\mu \eta (Z)hY-\nu \eta (Z)\phi {h}Y,U)\nonumber \\&+\sigma (-\kappa \eta (Z)Y-\mu \eta (Z)hY-\nu \eta (Z)\phi {h}Y,\phi {h}U)\nonumber \\= & {} \eta (Z)\{\kappa \sigma (Y,U)+\mu \sigma (hY,U)+\nu \phi \sigma ({h}Y,U)\nonumber \\&-\kappa \sigma (Y,\phi {h}U)+\mu (\kappa +1)\phi \sigma (U,Y)\nonumber \\&-\nu (\kappa +1)\sigma (U,Y)\}. \end{aligned}$$

(51)

Consequently, the values of (45–51) are put in (44), we have

$$\begin{aligned}&L_2\{\eta (U)\sigma (Y,Z)-\eta (U)\phi \sigma (hY,Z)-\eta (Y)\sigma (U,Z)\nonumber \\&\qquad +\eta (Y)\phi \sigma (hU,Z)+\eta (Z)\sigma (U,Y)-\eta (Z)\phi \sigma (hU,Y)\nonumber \\&\qquad -(\widetilde{\nabla }_U\sigma )(Y,Z)\}\nonumber \\&\quad =R^\bot (\xi ,Y)\sigma (U,Z)-R^\bot (\xi ,Y)\phi \sigma ({h}U,Z)\nonumber \\&\qquad +\eta (U)\{\kappa \sigma (Y,Z)+\mu \sigma (hY,Z)+\nu \phi \sigma (hY,Z)\nonumber \\&\qquad -\kappa \phi \sigma (hY,Z)+\mu (\kappa +1)\phi \sigma (Y,Z)\nonumber \\&\qquad +\nu (\kappa +1)\sigma (Y,Z)\}+\eta (Z)\{\kappa \sigma (Y,U)+\mu \sigma (hY,U)\nonumber \\&\qquad +\nu \phi \sigma (hY,U)-\kappa \phi \sigma (Y,hU)+\mu (\kappa +1)\phi \sigma (U,Y)\nonumber \\&\qquad -\nu (\kappa +1)\sigma (U,Y)\}\nonumber \\&\qquad +(\widetilde{\nabla }_U\sigma )(\kappa [\eta (Y)\xi -Y]-\mu {h}Y-\nu \phi {h}Y,Z). \end{aligned}$$

(52)

Taking $Z=\xi$ in (52) and taking into account Proposition 2.1, (52) reduce to

$$\begin{aligned}&L_2\{\sigma (U,Y)-\phi \sigma (hU,Y)-(\widetilde{\nabla }_U\sigma )(Y,\xi )\}=\kappa \sigma (Y,U)+\mu \sigma (hY,U)\nonumber \\&\quad +\nu \phi \sigma (hY,U)-\kappa \phi \sigma (Y,hU)+\mu (\kappa +1)\phi \sigma (U,Y)\nonumber \\&\quad -\nu (\kappa +1)\sigma (U,Y)\nonumber \\&\quad +(\widetilde{\nabla }_U\sigma )(\kappa [\eta (Y)\xi -Y]-\mu {h}Y-\nu \phi {h}Y,\xi ), \end{aligned}$$

(53)

where, by direct calculations, one can easily see that

$$\begin{aligned}&(\widetilde{\nabla }_U\sigma )(\kappa [\eta (Y)\xi -Y]-\mu {h}Y-\nu \phi {h}Y,\xi )\nonumber \\&\quad =-\sigma (\nabla _U\xi ,\kappa [\eta (Y)\xi -Y]-\mu {h}Y-\nu \phi {h}Y)\nonumber \\&\quad =\sigma (\phi ^2U+\phi {h}U,\kappa [\eta (Y)\xi -Y]-\mu {h}Y-\nu \phi {h}Y)\nonumber \\&\quad =-\sigma (U,\kappa [\eta (Y)\xi -Y]-\mu {h}Y-\nu \phi {h}Y)\nonumber \\&\qquad +\sigma (\phi {h}U,\kappa [\eta (Y)\xi -Y]-\mu {h}Y-\nu \phi {h}Y)\nonumber \\&\quad =\kappa \sigma (Y,U)+\mu \sigma (hY,U)+\nu \phi \sigma (U,hY)-\kappa \phi \sigma (Y,hU)\nonumber \\&\qquad +\mu (\kappa +1)\phi \sigma (U,Y)-\nu (\kappa +1)\sigma (U,Y) \end{aligned}$$

(54)

and

$$\begin{aligned} (\widetilde{\nabla }_U\sigma )(Y,\xi )= & {} -\sigma (\nabla _U\xi ,Y)=\sigma (\phi ^2U+\phi {h}U,Y)\nonumber \\= & {} \phi \sigma (hU,Y)-\sigma (U,Y). \end{aligned}$$

(55)

If (54) and (55) are put in (53), we reach at

$$\begin{aligned}{}[L_2-\kappa +(\nu -\mu \phi )(\kappa +1)]\sigma (U,Y)&-[(L_2-\kappa )\phi +(\mu +\phi \nu )]\sigma (U,hY)\nonumber \\= & {} 0. \end{aligned}$$

(56)

Substituting hY instead of Y in (56), by virtue of Proposition 2.1 and (6), we can easily see that

$$\begin{aligned} (\kappa +1)[(L_2-\kappa )\phi +(\mu +\phi \nu )]\sigma (U,Y)&+[L_2-\kappa +(\nu -\mu \phi )(\kappa +1)]\sigma (U,hY)\nonumber \\= & {} 0. \end{aligned}$$

(57)

From common solutions of (56) and (57) provided $\kappa \ne {0}$, we can infer

$$\begin{aligned}&[(L_2-\kappa )^2-(\kappa +1)(\nu ^2-\mu ^2)]\sigma (U,Y)\nonumber \\&\quad +2\mu \nu (\kappa +1)\phi \sigma (U,Y)=0. \end{aligned}$$

(58)

This implies that M is either totally geodesic or (42) is satisfied. Thus, the proof is completed. $\square$

From Theorem 2.6, we have the following corollary.

Corollary 2.7

Let M be an invariant submanifold of an almost Kenmotsu $(\kappa ,\mu ,\nu )$-space $\widetilde{M}^{2n+1}(\phi ,\eta ,\xi ,g)$. M is 2-semiparallel submanifold if and only if M is totally geodesic submanifold provided

$$\begin{aligned} \kappa ^2-(\kappa +1)(\nu ^2-\mu ^2)\ne {0}, \ \ or \ \ \mu \nu (\kappa +1)\ne {0}. \end{aligned}$$

Theorem 2.8

Let M be an invariant submanifold of an almost Kenmotsu $(\kappa ,\mu ,\nu )$-space $\widetilde{M}^{2n+1}(\phi ,\eta ,\xi ,g)$. If M is a 2-generalized Ricci pseudoparallel submanifold of $\widetilde{M}^{2n+1}(\phi ,\eta ,\xi ,g)$, then M is either totally geodesic or the function $L_4$ satisfies the condition

$$\begin{aligned} L_4=\frac{1}{2n}\left( 1\mp \frac{1}{k}\sqrt{(\kappa +1)(\nu ^2-\mu ^2)}\right) , \ \ \mu .\nu .(\kappa +1)=0. \end{aligned}$$

(59)

Proof

Let M be an invariant 2-generalized Ricci pseudoparallel submanifold of an almost Kenmotsu $\widetilde{M}^{2n+1}(\phi ,\eta ,\xi ,g)$-space. It follows that

$$\begin{aligned} L_4Q(S,\widetilde{\nabla }\sigma )(U,V,Z;X,Y)=(\widetilde{R}(X,Y)\cdot \widetilde{\nabla }\sigma )(U,V,Z), \end{aligned}$$

(60)

for all $X,Y,Z,U,V\in \Gamma (TM)$, that is,

$$\begin{aligned}&-L_4\{(\widetilde{\nabla }_{(X\wedge _SY)U}\sigma )(V,Z)+(\widetilde{\nabla }_U\sigma )((X\wedge _SY)V,Z)\nonumber \\&\quad +(\widetilde{\nabla }_U\sigma )(V,(X\wedge _SY)Z)\}=R(X,Y)(\widetilde{\nabla }_U\sigma )(V,Z)\nonumber \\&\quad -(\widetilde{\nabla }_{R(X,Y)U}\sigma )(V,Z)-(\widetilde{\nabla }_U\sigma )(R(X,Y)V,Z)\nonumber \\&\quad -(\widetilde{\nabla }_U\sigma )(V,R(X,Y)Z). \end{aligned}$$

This equality reduces for $X=V=\xi$

$$\begin{aligned}&-L_4\{(\widetilde{\nabla }_{(\xi \wedge _SY)U}\sigma )(\xi ,Z)+(\widetilde{\nabla }_U\sigma )((\xi \wedge _SY)\xi ,Z)\nonumber \\&\quad +(\widetilde{\nabla }_U\sigma )(\xi ,(\xi \wedge _SY)Z)\}=R(\xi ,Y)(\widetilde{\nabla }_U\sigma )(\xi ,Z)\nonumber \\&\quad -(\widetilde{\nabla }_{R(\xi ,Y)U}\sigma )(\xi ,Z)-(\widetilde{\nabla }_U\sigma )(R(\xi ,Y)\xi ,Z)\nonumber \\&\quad -(\widetilde{\nabla }_U\sigma )(\xi ,R(\xi ,Y)Z). \end{aligned}$$

(61)

Now, we will calculate these expressions separately. Firstly, making use of (6), (10), (15) and (29), we have

$$\begin{aligned} (\widetilde{\nabla }_{(\xi \wedge _SY)U}\sigma )(\xi ,Z)= & {} -\sigma (\nabla _{(\xi \wedge _SY)U}\xi ,Z)=\sigma (\phi ^2(\xi \wedge _SY)U,Z)\nonumber \\&+\sigma (\phi {h}(\xi \wedge _SY)U,Z)\nonumber \\= & {} -\sigma (S(Y,U)\xi -S(\xi ,U)Y,Z)\nonumber \\&+\sigma (\phi {h}(S(Y,U)\xi -S(\xi ,U)Y),Z)\nonumber \\= & {} 2n\kappa \eta (U)\{\sigma (Y,Z)-\phi \sigma (hY,Z)\}, \end{aligned}$$

(62)

$$\begin{aligned} (\widetilde{\nabla }_U\sigma )((\xi \wedge _SY)\xi ,Z)= & {} (\widetilde{\nabla }_U\sigma )(S(Y,\xi )\xi -S(\xi ,\xi )Y,Z)\nonumber \\= & {} 2n\{(\widetilde{\nabla }_U\sigma )(\kappa \eta (Y)\xi ,Z)-(\widetilde{\nabla }_U\sigma )(\kappa {Y},Z)\}\nonumber \\= & {} 2n\{-\sigma (\nabla _U\kappa \eta (Y)\xi ,Z)-(\widetilde{\nabla }_U\sigma )(\kappa {Y},Z)\}\nonumber \\= & {} 2n\{-\sigma (U\eta [\kappa (Y)]\xi +\kappa \eta (Y)\nabla _U\xi ,Z)\nonumber \\&-(\widetilde{\nabla }_U\sigma )(\kappa {Y},Z)\}\nonumber \\= & {} 2n\{\kappa \eta (Y)\sigma (\phi ^2U+\phi {h}U,Z)-(\widetilde{\nabla }_U\sigma )(\kappa {Y},Z)\}\nonumber \\= & {} 2n\eta (Y)\kappa \{\phi \sigma (hY,U)-\sigma (U,Z)\}\nonumber \\&-2n(\widetilde{\nabla }_U\sigma )(\kappa {Y},Z), \end{aligned}$$

(63)

$$\begin{aligned} (\widetilde{\nabla }_U\sigma )(\xi ,S(Y,Z)\xi -S(\xi ,Z)Y)= & {} (\widetilde{\nabla }_U\sigma )(\xi ,S(Y,Z)\xi )\nonumber \\&-2n(\widetilde{\nabla }_U\sigma )(\kappa \eta (Z)Y,\xi )\nonumber \\= & {} -2n(\widetilde{\nabla }_U\sigma )(\kappa \eta (Z)Y,\xi )\nonumber \\= & {} 2n\kappa \sigma (\nabla _U\xi ,\eta (Z)Y)\nonumber \\= & {} -2n\kappa \sigma (\phi ^2U+\phi {h}U,\eta (Z)Y)\nonumber \\= & {} 2n\kappa \eta (Z)\{\sigma (U,Y)-\sigma (\phi {h}U,Y)\}. \end{aligned}$$

(64)

Next, let’s calculate the right side of the equality.

$$\begin{aligned} R^\bot (\xi ,Y)(\widetilde{\nabla }_U\sigma )(\xi ,Z)= & {} -R^\bot (\xi ,Y)\sigma (\nabla _U\xi ,Z)=R^\bot (\xi ,Y)\sigma (\phi ^2U+\phi {h}U,Z)\nonumber \\= & {} R^\bot (\xi ,Y)\{\phi \sigma (hU,Z)-\sigma (U,Z)\}. \end{aligned}$$

(65)

Furthermore, by virtue of Proposition 2.1, (6) and (11), we observe

$$\begin{aligned} (\widetilde{\nabla }_{R(\xi ,Y)U}\sigma )(\xi ,Z)= & {} -\sigma (\nabla _{R(\xi ,Y)U}\xi ,Z)=\sigma (\phi ^2R(\xi ,Y)U,Z)\nonumber \\&+\sigma (\phi {h}R(\xi ,Y)U,Z)=-\sigma (R(\xi ,Y)U,Z)\nonumber \\&+\sigma (\phi {h}R(\xi ,Y)U,Z)\nonumber \\= & {} \eta (U)\{\kappa \sigma (Y,Z)+\mu \sigma (hY,Z)+\nu \sigma (\phi {h}U,Z)\nonumber \\&-\kappa \sigma (\phi {h}Y,Z)-\mu \sigma (\phi {h}^2Y,Z)-\nu \sigma (\phi {h}\phi {h}Y,Z)\}\nonumber \\= & {} \eta (U)\{\kappa \sigma (Y,Z)+\mu \sigma (hY,Z)+\nu \sigma (\phi {h}Y,Z)\nonumber \\&-\kappa \sigma (\phi {h}Y,Z)+\mu (\kappa +1)\phi \sigma (Y,Z)\nonumber \\&+\nu (\kappa +1)\sigma (Y,Z)\}, \end{aligned}$$

(66)

$$\begin{aligned}&(\widetilde{\nabla }_U\sigma )(R(\xi ,Y)\xi ,Z)\nonumber \\&\quad =(\widetilde{\nabla }_U\sigma )(\kappa [\eta (Y)\xi -Y]-\mu {h}Y-\nu \phi {h}Y,Z). \end{aligned}$$

(67)

Finally,

$$\begin{aligned} (\widetilde{\nabla }_U\sigma )(\xi ,R(\xi ,Y)Z)= & {} -\sigma (\nabla _U\xi ,R(\xi ,Y)Z)\nonumber \\= & {} \sigma (\phi ^2U+\phi {h}U,R(\xi ,Y)Z)\nonumber \\= & {} -\sigma (U,R(\xi ,Y)Z)+\sigma (\phi {h}U,R(\xi ,Y)Z)\nonumber \\= & {} \eta (Z)\{\sigma (U,Y)+\mu \sigma (U,hY)+\nu \phi \sigma (U,hY)\nonumber \\&-\kappa \phi \sigma (hU,Y)-\mu \phi \sigma (hU,hY)-\nu \phi ^2\sigma (hY,hU)\}\nonumber \\= & {} \eta (Z)\{\kappa \sigma (U,Y)+\mu \sigma (hY,U)+\nu \phi \sigma (U,hY)\nonumber \\&-\kappa \phi \sigma (hU,Y)+\mu (\kappa +1)\phi \sigma (U,Y)\nonumber \\&-\nu (\kappa +1)\sigma (U,Y)\} \end{aligned}$$

(68)

If (62) and (68) are put in (61), we have

$$\begin{aligned}&2n\kappa {L_4}\{\eta (U)\sigma (Y,Z)-\eta (U)\phi \sigma (hY,Z)-\eta (Y)\sigma (U,Z)+\eta (Y)\phi \sigma (hU,Z)\nonumber \\&\qquad +\eta (Z)\sigma (U,Y)-\eta (Z)\phi \sigma (hU,Y)-(\widetilde{\nabla }_U\sigma )(Y,Z)\}\nonumber \\&\quad =R^\bot (\xi ,Y)\{\sigma (U,Z)-\phi \sigma (hU,Z)\}+\eta (U)\{\kappa \sigma (Y,Z)\nonumber \\&\qquad +\mu \sigma (hY,Z)+\nu \phi \sigma (hY,Z)-\kappa \phi \sigma (hY,Z)\nonumber \\&\qquad +\mu (\kappa +1)\phi \sigma (Y,Z)\}+\eta (Z)\{\kappa \sigma (U,Y)+\mu \sigma (U,hY)\nonumber \\&\qquad +\nu \phi \sigma (U,hY)-\kappa \phi \sigma (hU,Y)+\mu (\kappa +1)\phi \sigma (U,Y)\nonumber \\&\qquad -\nu (\kappa +1)\sigma (U,Y)\}\nonumber \\&\qquad +(\widetilde{\nabla }_U\sigma )(\kappa [\eta (Y)\xi -Y]-\mu {h}Y-\nu \phi {h}Y,Z). \end{aligned}$$

(69)

Taking $Z=\xi$ in (69), we arrive at

$$\begin{aligned}&2n\kappa {L_4}\{\sigma (U,Y)-\phi \sigma (hU,Y)-(\widetilde{\nabla }_U\sigma )(Y,\xi )\}=\kappa \sigma (U,Y)+\mu \sigma (U,hY)\nonumber \\&\quad +\phi \sigma (U,hY)-\kappa \phi \sigma (hU,Y)+\mu (\kappa +1)\sigma (U,Y)\nonumber \\&\quad +(\widetilde{\nabla }_U\sigma )(\kappa [\eta (Y)\xi -Y]-\mu {h}Y-\nu \phi {h}Y,\xi )\nonumber \\&\quad -\nu (\kappa +1)\sigma (U,Y), \end{aligned}$$

(70)

where

$$\begin{aligned}&(\widetilde{\nabla }_U\sigma )(\kappa [\eta (Y)\xi -Y]-\mu {h}Y-\nu \phi {h}Y,\xi )\nonumber \\&\quad =-\sigma (\nabla _U\xi ,\kappa [\eta (Y)\xi -Y]-\mu {h}Y-\nu \phi {h}Y)\nonumber \\&\quad =\sigma (\phi ^2U+\phi {h}U,\kappa [\eta (Y)\xi -Y]-\mu {h}Y-\nu \phi {h}Y)\nonumber \\&\quad =\kappa \sigma (U,Y)+\mu \sigma (U,hY)+\nu \phi \sigma (U,hY)-\kappa \phi \sigma (Y,hU)\nonumber \\&\qquad +\mu (\kappa +1)\phi \sigma (U,Y)-\nu (\kappa +1)\sigma (U,Y) \end{aligned}$$

(71)

and

$$\begin{aligned} (\widetilde{\nabla }_U\sigma )(Y,\xi )= & {} -\sigma (\nabla _U\xi ,Y)=\sigma (\phi ^2U+\phi {h}U,Y)\nonumber \\= & {} \phi \sigma (hU,Y)-\sigma (U,Y). \end{aligned}$$

(72)

From (69), (70) and (71), we conclude

$$\begin{aligned} 2n\kappa {L_4}\{\sigma (U,Y)-\phi \sigma (hU,Y)\}= & {} \kappa \sigma (U,Y)+\mu \sigma (U,hY)+\nu \phi \sigma (U,hY)\nonumber \\&-\kappa \phi \sigma (hU,Y)+\mu (\kappa +1)\phi \sigma (U,Y)\nonumber \\&-\nu (\kappa +1)\sigma (U,Y), \end{aligned}$$

that is,

$$\begin{aligned}{}[2n\kappa {L_4}-\kappa +\nu (\kappa +1)]\sigma (U,Y)= & {} \mu (\kappa +1)\phi \sigma (U,Y)+\mu \sigma (U,hY)\nonumber \\&+[2n\kappa {L_4}+\nu -\kappa ]\phi \sigma (U,hY). \end{aligned}$$

(73)

Substituting hY for Y in (73) and again considering (6) and (11), we conclude

$$\begin{aligned}&[2n\kappa {L_4}-\kappa +\nu (\kappa +1)]\sigma (U,hY)=-(\kappa +1)[2n\kappa {L_4}+\nu -\kappa ]\phi \sigma (U,Y)\nonumber \\&\quad +\mu (\kappa +1)\phi \sigma (U,hY)-\mu (\kappa +1)\sigma (U,Y). \end{aligned}$$

(74)

From (73) and (74), we observe

$$\begin{aligned}{}[\kappa ^2(2nL_4-1)^2-(\kappa +1)(\nu ^2-\mu ^2)]\sigma (U,Y)+2\mu \nu (\kappa +1)\phi \sigma (U,Y)=0. \end{aligned}$$

Since $\sigma$ and $\phi \sigma$ are orthogonal, the last equality implies that $\sigma$ is either identically zero or (59) is satisfied. $\square$